Stability-Certified Koopman Observer Design for Nonlinear Systems via Generalized Persidskii Dynamics

Technical Summary: Stability-Certified Koopman Observer Design for Nonlinear Systems via Generalized Persidskii Dynamics

Problem Statement
The paper addresses the challenge of nonlinear state estimation for dynamical systems where the governing equations are approximated using Koopman operator liftings. While Koopman-based predictors (often realized via Extended Dynamic Mode Decomposition, or EDMD) offer powerful data-driven linear representations of nonlinear dynamics, they suffer from a fundamental limitation: the "lifting residual." This residual represents the portion of the nonlinear vector field not captured by the finite-dimensional observable dictionary.

Existing approaches to observer design in this context typically rely on empirical regularization or local linearization arguments valid only near fixed points. Consequently, there is a lack of computable, global convergence certificates for Koopman-based observers, particularly when facing model mismatch (large lifting residuals) and external measurement noise. The central problem is to design an observer that guarantees stability and convergence despite these unavoidable approximation errors.

Methodology
The authors propose a novel observer design that bridges Koopman theory with the stability framework of generalized Persidskii systems. The methodology proceeds through the following steps:

1. Error Dynamics Formulation: The authors define the estimation error in the lifted latent space. They decompose the total error source into a "lifting residual" and measurement noise.
2. Structural Correspondence: A key theoretical insight is establishing that the error dynamics of a Koopman latent-space observer, when equipped with a specific nonlinear correction term, can be exactly reformulated as a generalized Persidskii system. This class of systems is characterized by nonlinearities satisfying uniform sector bounds and admits diagonal Lyapunov functions.
3. Residual Decomposition: The lifting residual is decomposed into two parts: a sector-admissible component (which fits the Persidskii structure) and a bounded unstructured remainder. This allows the observer to explicitly account for the structured part of the approximation error.
4. LMI-Based Gain Synthesis: The observer gain is computed by solving a Linear Matrix Inequality (LMI). This LMI is derived from a diagonal Lyapunov function candidate and ensures Input-to-State Stability (ISS) of the estimation error with respect to both the lifting residuals and external disturbances.
5. Stability Analysis: The design guarantees exponential convergence in the ideal case (zero residual/noise) and ultimate boundedness in the perturbed case, with an explicit bound on the steady-state error that scales linearly with the magnitude of the lifting residual.

Key Contributions
The paper makes four specific contributions:

• Characterization of Reformulation: It provides a precise characterization of the conditions under which Koopman error dynamics admit a generalized Persidskii reformulation, including a constructive procedure for decomposing the lifting residual.
• ISS-Certified Observer Design: It derives an LMI-based observer gain that simultaneously minimizes the ISS gain from the residual to the estimation error, providing a computable certificate of robustness.
• Convergence Proofs: It analytically proves exponential convergence for the nominal case and establishes ultimate boundedness with an explicit error bound for the case of persistent bounded perturbations.
• Numerical Validation: It validates the approach on two benchmark systems, demonstrating superior performance over existing methods.

Experimental Results
The proposed "Persidskii-Koopman Observer" (PKO) was tested against a standard Extended Kalman Filter (EKF) and a linear Koopman observer (Lin-Koop) on two benchmarks:

1. Van der Pol Oscillator: Under Gaussian noise and a 15% parameter mismatch, the PKO maintained tight tracking of the limit cycle, whereas the Lin-Koop observer accumulated errors near switching regions and the EKF exhibited intermittent divergence.
2. Nonlinear Robotic Arm: A single-link arm with significant friction uncertainty (±30%) and measurement noise was used. The PKO tracked the angular velocity closely, while the EKF and Lin-Koop showed persistent amplitude bias.

Quantitative results across 100 trials showed that the PKO achieved a 42% reduction in steady-state Root Mean Square Error (RMSE) compared to the EKF and a 34–35% reduction compared to the linear Koopman observer. The variance of the PKO was also consistently lower, indicating improved robustness. The computational cost of solving the LMI was negligible (under 0.65 seconds on a standard workstation), confirming the practical tractability of the approach.

Significance and Claims
The paper claims to resolve the open problem of certifying observer convergence for Koopman-based predictors under model mismatch. By leveraging the structural properties of generalized Persidskii systems, the authors provide a rigorous, global stability certificate that is absent in current data-driven observer designs.

The significance lies in the ability to design observers that are not only data-driven but also robust to the inherent limitations of finite-dimensional Koopman approximations. The authors emphasize that their method offers a "provably ISS" design with an explicit ultimate bound, outperforming both traditional nonlinear filters (EKF) and linearized Koopman approaches without increasing computational complexity beyond a one-time offline optimization. The work suggests that future extensions could apply this diagonal Lyapunov structure to multi-agent systems and time-varying delays, building upon the established connection between Koopman dynamics and Persidskii stability theory.